Set 1

TipInstructions
  • Here are some additional practice questions and activities for you to work on independently.
  • Some of these questions are from previous exams and tests.
  • I will not provide written solutions, as I want to avoid them being circulated to future batches, which would render the practice less effective. However, I am more than happy to discuss and assist you as you work through them.
    Please feel free to email me at (chammika[at]nus.edu.sg) to organise a meeting.

Problems

Problem 1.1(Relativistic Relations)

Starting with the definition for the energy-momentum four vector, use Equation 2.5 and Equation 2.6 to arrive at Equation 2.8.

Problem 1.2(Decay of a \(\rho\) Meson)

\[ \rho \to \pi^0 + \gamma \]

For the above disintegration, calculate the energy of the photon and the kinetic energy of the pion in the rest frame of the \(\rho\) meson.

Take the mass of \(\rho\) as 770 MeV and \(\pi^0\) as 135 MeV, respectively.

Problem 1.3(Pion Decay at Rest)

A pion at rest, in the Lab reference frame, decays via \[ \pi\rightarrow\mu+\nu. \tag{1.1}\]

  1. Improve and rewrite the above equation by adding more information about the particles.
  2. Show that the speed of the muon is given by: \[ v=\left(\frac{m_\pi^2-m_\mu^2}{m_\pi^2+m_\mu^2}\right)c. \tag{1.2}\] Please justify and assumptions that you need to make.
  3. If you were to detect the emitted muon, in which direction would you place your detector? Is there a preferred direction?

Problem 1.4(The Decay of \(\pi^-\))

The following equation shows a decay of a zero spin pion:

\[ \me{\pi^-}{\mu^-+\Amun} \]

Discuss the impact of the orbital angular momentum on the alignment of the spins of the muon and neutrino.

Problem 1.5(Mandelstam Variables)

The Mandelstam variables \(s\), \(t\), and \(u\) are defined for the reaction \[ A+B\rightarrow C+D \tag{1.3}\]

by

\[ s=\frac{(p^\mu_A+p^\mu_B)^2}{c^2}, \qquad t=\frac{(p^\mu_A-p^\mu_C)^2}{c^2}, \qquad u=\frac{(p^\mu_A-p^\mu_D)^2}{c^2}, \tag{1.4}\]

where \(p^\mu_A\), etc. are the relevant energy–momentum four-vectors.

  1. Show that

    \[ s+t+u=\sum_{j=A,B,C,D}m_j^2. \tag{1.5}\]

  2. In the case of elastic scattering, show that

    \[ t=-\frac{2p^2}{c^2}(1-\cos\theta), \tag{1.6}\]

    where \(p\equiv|\mathbf{p}|\), \(\mathbf{p}\) is the centre-of-mass momentum of particle \(A\), and \(\theta\) is its scattering angle in the centre-of-mass frame.

Problem 1.6(Quark Spin and the Pauli Principle)

  1. A proton has the quark content \(uud\). By combining the spins of the three quarks, determine the possible values of the total quark spin \(S\).

  2. The proton is observed to have a total spin of \(J=\frac{1}{2}\). Assuming that the quarks are in their ground state, so that the orbital angular momentum is \(L=0\), which of the possible values of \(S\) corresponds to the proton?

  3. Now consider the observed \(\Delta^{++}\) baryon, which has the quark content \(uuu\) and spin \(J=\frac{3}{2}\). In its ground state, the orbital angular momentum is \(L=0\).

    1. Determine, with justification, the exchange symmetry of the spatial, flavour, and spin parts of the \(\Delta^{++}\) wavefunction.

    2. Hence, determine the exchange symmetry of the combined spatial–spin–flavour wavefunction. Why does this appear to contradict the Pauli exclusion principle?

  4. The existence of the \(\Delta^{++}\) suggests that quarks possess an additional quantum number called colour. What exchange symmetry must the colour part of the wavefunction possess so that the complete \(\Delta^{++}\) wavefunction satisfies the Pauli exclusion principle?